Fake violations of the quantum Bell-parameter bound

Shortcomings of experimental techniques are usually assumed to diminish nonclassical properties of quantum systems. Here it is demonstrated that this standard assumption is not true in general. It is theoretically shown that the inability to resolve different photon numbers with photodetection may pseudoincrease a measured Bell parameter. Under proper conditions one even pseudoviolates the quantum Cirel'son bound of the Bell parameter, and the corresponding density operator fails to be positive semidefinite. This paradox can be resolved by appropriate squash models.